The contractivity- preserving 2- and 3- step predictor- corrector series methods for ODEs are expanded into new optimal, contractivity-preserving, d-derivative, k-step, predictor-corrector HBO series methods, denoted by HBO(d,k,p) woth nonnegative coefficients for solving nonstiff first-order initial value problems. The main reason for considering this class of formulae is to obtain a set of methods which have larger regions of stability and generallyhigher upper bound of order p of HBO(d,k,p) for given d. Their stability regions have generally a good shape and grow generally with decreasing p-d. A selected CP HBO method: 6-derivative 4-step HBO of order 14, denoted by HBO(6,4,14) which has a maximum order 14 based on the CP conditions compares satisfactorily with Adams-Cowell of order 13 in PECE mode, denoted by AC(13), in solving as a function of the CPU time. HBO(6,4,14) also compares well with AC(13) in solving standard N-body problems on the basis of the growth of relative, positional error, relative energy error and 1000 periods of integration.